Properties

Label 475.2.a.e
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Defining polynomial: \(x^{3} - x^{2} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{6} + ( -1 + \beta_{2} ) q^{7} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{8} + \beta_{1} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{6} + ( -1 + \beta_{2} ) q^{7} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{8} + \beta_{1} q^{9} -\beta_{2} q^{11} + ( -3 + \beta_{1} - \beta_{2} ) q^{12} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{13} + ( 2 - \beta_{2} ) q^{14} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{16} + ( -5 - \beta_{2} ) q^{17} + ( 3 + \beta_{2} ) q^{18} + q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{21} + ( -1 - \beta_{1} + \beta_{2} ) q^{22} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{23} + 5 q^{24} + ( -7 + 4 \beta_{1} - 6 \beta_{2} ) q^{26} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} ) q^{28} + ( -2 - \beta_{2} ) q^{29} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{31} + ( -2 \beta_{1} + \beta_{2} ) q^{32} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( 4 - 6 \beta_{1} + \beta_{2} ) q^{34} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{36} + ( -4 + 2 \beta_{1} - 5 \beta_{2} ) q^{37} + ( -1 + \beta_{1} ) q^{38} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{39} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{41} + ( \beta_{1} - 3 \beta_{2} ) q^{42} + ( -1 + 6 \beta_{1} - 2 \beta_{2} ) q^{43} - q^{44} + ( -6 + \beta_{1} - 5 \beta_{2} ) q^{46} + ( -5 + 5 \beta_{1} - \beta_{2} ) q^{47} + ( 1 + 3 \beta_{1} + 2 \beta_{2} ) q^{48} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{49} + ( 7 + \beta_{1} + 4 \beta_{2} ) q^{51} + ( 11 - 7 \beta_{1} + 4 \beta_{2} ) q^{52} + ( -9 - 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -5 + 5 \beta_{1} - 5 \beta_{2} ) q^{54} + ( -1 - 2 \beta_{1} + 4 \beta_{2} ) q^{56} + ( -1 - \beta_{2} ) q^{57} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{58} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} ) q^{61} + ( -1 + 4 \beta_{1} - 4 \beta_{2} ) q^{62} + q^{63} + ( -3 + 5 \beta_{1} - 5 \beta_{2} ) q^{64} + ( \beta_{1} + 2 \beta_{2} ) q^{66} + ( 6 - 4 \beta_{1} + \beta_{2} ) q^{67} + ( -11 + 5 \beta_{1} - 5 \beta_{2} ) q^{68} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{69} + ( 3 - 2 \beta_{1} ) q^{71} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{72} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 5 - 9 \beta_{1} + 7 \beta_{2} ) q^{74} + ( 2 - \beta_{1} + \beta_{2} ) q^{76} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{77} + ( 15 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( -3 - 4 \beta_{1} + 5 \beta_{2} ) q^{79} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{81} + ( 15 - 4 \beta_{1} + 7 \beta_{2} ) q^{82} + ( -2 - 3 \beta_{1} - 6 \beta_{2} ) q^{83} + ( 2 - \beta_{1} ) q^{84} + ( 17 - 3 \beta_{1} + 8 \beta_{2} ) q^{86} + ( 4 + \beta_{1} + \beta_{2} ) q^{87} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{88} + ( -9 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 + 3 \beta_{1} - 8 \beta_{2} ) q^{91} + ( 6 - 5 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -6 - \beta_{1} + 2 \beta_{2} ) q^{93} + ( 19 - 6 \beta_{1} + 6 \beta_{2} ) q^{94} + ( 3 \beta_{1} + \beta_{2} ) q^{96} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{97} + ( 3 - 8 \beta_{1} + 5 \beta_{2} ) q^{98} + ( -1 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + O(q^{10}) \) \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24} - 11 q^{26} + q^{27} - q^{28} - 5 q^{29} - q^{31} - 3 q^{32} + 8 q^{33} + 5 q^{34} - 3 q^{36} - 5 q^{37} - 2 q^{38} - 11 q^{39} + q^{41} + 4 q^{42} + 5 q^{43} - 3 q^{44} - 12 q^{46} - 9 q^{47} + 4 q^{48} - 7 q^{49} + 18 q^{51} + 22 q^{52} - 31 q^{53} - 5 q^{54} - 9 q^{56} - 2 q^{57} - q^{58} - 6 q^{59} + 3 q^{61} + 5 q^{62} + 3 q^{63} + q^{64} - q^{66} + 13 q^{67} - 23 q^{68} + q^{69} + 7 q^{71} - q^{72} - q^{73} - q^{74} + 4 q^{76} - 10 q^{77} + 42 q^{78} - 18 q^{79} - 21 q^{81} + 34 q^{82} - 3 q^{83} + 5 q^{84} + 40 q^{86} + 12 q^{87} + 12 q^{88} - 20 q^{89} + 17 q^{91} + 11 q^{92} - 21 q^{93} + 45 q^{94} + 2 q^{96} + 13 q^{97} - 4 q^{98} - 4 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.37720
−0.273891
2.65109
−2.37720 −1.27389 3.65109 0 3.02830 −0.726109 −3.92498 −1.37720 0
1.2 −1.27389 1.65109 −0.377203 0 −2.10331 −3.65109 3.02830 −0.273891 0
1.3 1.65109 −2.37720 0.726109 0 −3.92498 0.377203 −2.10331 2.65109 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.a.e 3
3.b odd 2 1 4275.2.a.bm 3
4.b odd 2 1 7600.2.a.cc 3
5.b even 2 1 475.2.a.g yes 3
5.c odd 4 2 475.2.b.b 6
15.d odd 2 1 4275.2.a.ba 3
19.b odd 2 1 9025.2.a.bc 3
20.d odd 2 1 7600.2.a.bh 3
95.d odd 2 1 9025.2.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.e 3 1.a even 1 1 trivial
475.2.a.g yes 3 5.b even 2 1
475.2.b.b 6 5.c odd 4 2
4275.2.a.ba 3 15.d odd 2 1
4275.2.a.bm 3 3.b odd 2 1
7600.2.a.bh 3 20.d odd 2 1
7600.2.a.cc 3 4.b odd 2 1
9025.2.a.y 3 95.d odd 2 1
9025.2.a.bc 3 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2 T_{2}^{2} - 3 T_{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - 3 T + 2 T^{2} + T^{3} \)
$3$ \( -5 - 3 T + 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -1 + T + 4 T^{2} + T^{3} \)
$11$ \( -1 - 4 T - T^{2} + T^{3} \)
$13$ \( -103 - 36 T + 3 T^{2} + T^{3} \)
$17$ \( 79 + 61 T + 14 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -125 - 9 T + 8 T^{2} + T^{3} \)
$29$ \( -5 + 4 T + 5 T^{2} + T^{3} \)
$31$ \( 53 - 30 T + T^{2} + T^{3} \)
$37$ \( -395 - 74 T + 5 T^{2} + T^{3} \)
$41$ \( 155 - 82 T - T^{2} + T^{3} \)
$43$ \( 317 - 113 T - 5 T^{2} + T^{3} \)
$47$ \( -311 - 64 T + 9 T^{2} + T^{3} \)
$53$ \( 905 + 303 T + 31 T^{2} + T^{3} \)
$59$ \( -200 - 40 T + 6 T^{2} + T^{3} \)
$61$ \( -1 - 10 T - 3 T^{2} + T^{3} \)
$67$ \( 169 - 13 T^{2} + T^{3} \)
$71$ \( 47 - T - 7 T^{2} + T^{3} \)
$73$ \( 53 - 30 T + T^{2} + T^{3} \)
$79$ \( -395 + 17 T + 18 T^{2} + T^{3} \)
$83$ \( 131 - 270 T + 3 T^{2} + T^{3} \)
$89$ \( -125 + 51 T + 20 T^{2} + T^{3} \)
$97$ \( 169 - 13 T^{2} + T^{3} \)
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